466 ON THE BINOMIAL THEOREM, [533
But the method of varied multiplication may be applied to the demonstration of
a much more general theorem ; viz. we may use it to develope a product such as
(A — a) (A - A) (h — c) (h — d) (h — e),
according to a series of products
(A — a) (A - /3) (A — 7) (A - A) (A - e),
(h - a) (A - /3) (h - 7) (A - $),
(h - a) (h - /3) (h - 7),
(A - a) (h - /3),
(h - a),
1.
For this purpose, starting with
h — a = h — a + a — a,
we multiply by h — b, written first under the form h — ¡3 + /3 — b, and then under the
form h — a. + (a — b) ; we have thus
(h — a)(h — b) = (h — a) (h — /3)
+ (A- a) {(a -a)-f (/3 — A)}
+ 1 (a — a) (a — b),
and so on. It is easy to see that we may for instance write
(A — a) (A — b) (A — c) (A — d) (A — e)
= (A - a) (A — /3) (A — 7) (A — S) (A — e)
4- (A — a) (A — /3) (A — 7) (A — 8) I
+ (A - a) (A - /3) (A - 7)
+ (A — a) (A — /3)
+ A — a
+ 1
a, ft 7, 3,
.a, b, c, d,
'a, ft 7, 3
.a, b, c, d,
ft 7
.a, A, c, ft
' a , £
A, c, ft
[a, b, c, ft